Vol. 26, No. 3, 1968

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p-solvable doubly transitive permutation groups

Donald Steven Passman

Vol. 26 (1968), No. 3, 555–577

In this paper doubly transitive and 3/2-transitive permutation groups are classified under hypotheses somewhat weaker than solvability. We mention two examples.

Let 𝒮(pn) denote the group of semilinear transformations over GF(pn). The following combines a result of Huppert on solvable 2-transitive groups and a result of Zassenhaus on sharply 2-transitive groups.

Theorem I. Let G be a p-solvable doubly transitive permutation group with Op(G)1. Then deg G = pn for some n and we have one of the following: (i) G 𝒮(pn), (ii) G is solvable and pn = 32,52,72,112,232 or 34, or (iii) G is nonsolvable and pn = 112,192,292 or 592.

The second result reads better as a theorem on linear groups.

Theorem II. Let group G act faithfully on vector space V over GF(p) and let G act 1/2-transitively but not semiregularly on V#. If G is imprimitive as a linear group, then G is solvable and we have one of the following: (i) |V| = p2n for p2 and G is a specific group of order 4(pn 1), (ii) |V| = 34 and |G| = 32, or (iii) |V| = 26 and |G| = 18.

Mathematical Subject Classification
Primary: 20.20
Received: 27 June 1967
Published: 1 September 1968
Donald Steven Passman